Numerical analysis of shrinkage stresses in a mass concrete

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Numerical analysis of shrinkage stresses in a massconcreteHow‐Ji Chen a , Hsien‐Sheng Peng b & Yi‐Feng Chen b

a Department of Civil Engineering , National Chung‐Hsing University , Taichung, Taiwan,402, R.O.C. Phone: 886–4–22859390; Fax: 886–4–22859390; E-mail:b Department of Civil Engineering , National Chung‐Hsing University , Taichung, Taiwan,402, R.O.C.Published online: 04 Mar 2011.

To cite this article: How‐Ji Chen , Hsien‐Sheng Peng & Yi‐Feng Chen (2004) Numerical analysis of shrinkage stresses in amass concrete, Journal of the Chinese Institute of Engineers, 27:3, 357-365, DOI: 10.1080/02533839.2004.9670882

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Journal of the Chinese Institute of Engineers, Vol. 27, No. 3, pp. 357-365 (2004) 357

NUMERICAL ANALYSIS OF SHRINKAGE STRESSES IN A MASS

CONCRETE

How-Ji Chen*, Hsien-Sheng Peng, and Yi-Feng Chen

ABSTRACT

A numerical model based on an actual structure in Taiwan is proposed to inves-tigate the shrinkage behavior of mass concrete. A commercially available program,ANSYS, which treats concrete as a homogeneous and isotropic material, was used inthe investigation. The stresses are assumed to be within the elastic range. The shrink-age strains for various segments of mass concrete were evaluated according to cur-rently available prediction equations (ACI 209, 1992; CEB-FIP, 1991). By convert-ing these strain values into equivalent stresses based on Hook’s law, finite elementcomputations were then performed to predict the long-term shrinkage behavior of massconcrete.

Experimental verifications by nondestructive tests were carried out to confirmthe results of the numerical predictions. Comparisons show that the numerical pre-dictions are in agreement with the experimental results.

Key Words: mass concrete, shrinkage stresses, finite element, nondestructive test.

*Corresponding author. (Tel: 886-4-22859390; Fax: 886-4-22855610; ; Email: [email protected])

The authors are with the Department of Civil Engineering, Na-tional Chung-Hsing University, Taichung, Taiwan 402, R.O.C.

I. INTRODUCTION

After being used for a long time, cracks in massconcrete may be found on the surface of a structure.This is due to a change of volume caused by the dry-ing shrinkage of concrete (Mehta, 1986). The dryingshrinkage of mass concrete is mainly caused by theuneven distribution of humidity, which also leads todifferent shrinkage strains in different sections(Mindess and Young, 1981; Kim and Lee, 1998).Accurate evaluation of the amount of concrete shrink-age is very important for pre-stressed concrete de-sign and predicting cracking in mass concrete. In-vestigators have proposed equations to predict thedrying shrinkage strain of concrete (ACI 209, 1992;CEB-FIP, 1991). However, these equations do notyield the same results. Hence, engineers and research-ers have tried to learn how to predict the dryingshrinkage strain in mass concrete precisely, as wellas the effect of drying shrinkage upon mass concrete.

A concrete cylindrical container (10.4 m indiameter, 10.8 m in height and 2.5 m thick, built on a

3 m thick foundation plate), located at the Instituteof Nuclear Energy Research in Taiwan, was selectedas the prototype structure for measurements andanalyses of actual cracks and shrinkage. The numeri-cal model is established based on the real scale of themass concrete structure to permit analysis by theANSYS program. The drying shrinkage strain in dif-ferent sections of mass concrete structures was evalu-ated in accordance with various prediction equations.The calculated drying shrinkage strain is then trans-formed into equivalent stress according to the gen-eral Hook’s law, which is then applied to the massconcrete model. Strains corresponding to equivalentstress were used to simulate the drying shrinkagestrain. Finally, a nondestructive test, base on theTime-of-Flight Diffraction Technique (Lin and Su,1996; Lin et al., 1999), is carried out to measure thedepth of cracks and to confirm the results of the nu-merical predictions.

II. CONCEPT OF EQUIVALENT STRESS

Since the value of concrete strain cannot be con-sidered as an input, the drying shrinkage strain istransformed into an equivalent stress in numericalcalculations. It is assumed that the materials are ho-mogeneous and elastic; the relationship between the

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358 Journal of the Chinese Institute of Engineers, Vol. 27, No. 3 (2004)

drying shrinkage strain and the equivalent stress canbe derived according to the general Hook’s law.

The stress-strain formula of the general Hook’slaw is

εx + εy + εz = 1 – 2νE (σ x + σ y + σ z) (1)

where ν is Poisson’s ratio, E is elastic modulus, εand σ represent respectively the strain and stress; thesubscripts denote the directions of the Cartesian axes.

Suppose that the e lement would shr inkhomogeneously, so the strains and stresses on thethree axes are isotropic. Therefore, one has

⇒ σ = E1 – 2ν ε (2)

where ε and σ represent respectively the dryingshrinkage strain and equivalent stress.

Two different shape elements and a beam modelwere chosen for verifying the applicability of the con-cept of equivalent stress. By using an element (Solid

Brick 8 nodes 45 element) with a 2:1:3 aspect ratio(length, width and height), a linear spring (Beam 4element) is added at each node in three directions withstiffness approaching zero (see Fig. 1(a)). The mate-rial parameters were used according to the actual con-crete data, which are E=21.3 GPa, ν=0.2, and the dry-ing shrinkage strain was assumed 500 µ. The equiva-lent stress calculated by the conversion formula (2)is 17.75 MPa. Tri-axial uniform stress was placedonto the element for calculation. The results of dry-ing shrinkage strains in the X, Y, and Z-axis direc-tions are all 500 µ, the same as the assumed value.Another analysis performed used a tapered elementas shown in Fig. 1(b). All parameters used are thesame as for the element described above. Aftercalculation, the drying shrinkage strains in X, Y, andZ-axis directions can be determined to be 500 µ,respectively. The strain result of the finite elementanalysis shows that the concept of equivalent stressis feasible. However, the corresponding stress resultis not the real stress caused by shrinkage. The origi-nal equivalent stress must be subtracted from the re-sult after analysis. In order to verify the point, a beamframe (the numerical model as shown in Fig. 2) isanalyzed. Suppose that each element of this beamendures 800 µ shrinkage strain in the Z-axis direction.The comparison of theoretical and numerical solu-tions at three different places on this beam is listedin Table 1. Due to the boundary condition of the twofixed ends, the numerical solution of strain in theZ-axis direction approaches zero. We find that the

(a)

(b)

Fig. 1 3-D solid element

Table 1 Comparison of theoretical and numeri-cal solutions

Beam L/4 L/2 3L/4

Numerical17.06 17.06 17.06σz solution

(MPa) Theoretical17.06 17.06 17.06

solution

Numerical1.17E-18 -1.17E-19 -1.17E-18

solutionεz Theoretical0 0 0

solution

Fig. 2 Beam model

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H. J. Chen et al.: Numerical Analysis of Shrinkage Stresses in a Mass Concrete 359

theoretical solution is also zero. As to stress, aftersubtracting equivalent stress from the numericalsolution, the solution is as same as the theoretical one(shown in Table 1). The result verified the applica-bility of the concept of equivalent stress.

Shrinkage is not the same in every part of a massconcrete structure. Three prediction formulas, thespecification of ACI 209 (1992), CEB-FIP (1991) andthe formula proposed by Kim and Lee (1998), wereused to determine the amount of drying shrinkagestrain. Stress and strain caused by shrinkage can beobtained according to the equivalent stress concept.

III. RELIABILITY OF NUMERICALCALCULATION

The precision of numerical calculation dependson whether the result is convergent after calculation.Different mesh methods were used to test the preci-sion of the numerical calculation. A uniform distrib-uted stress P applied from inside radiates toward theoutside to simulate a container enduring an internalexploding stress. The numerical model is of a con-tainer with a circular wall and solid plate bottom (asshown in Fig. 3). The aspect ratio of element is keptunder 4 and the inside angle at each corner is between45° and 135°. An internal pressure of 50 MPa isapplied. Considering the actual boundary conditionof the container, the center of the bottom is fixed inthree directions (X, Y, and Z-axis) and the other partsof the bottom are only constrained in the Z direction,the circular wall is free. The material parameter iscompressive strength (fc’)=20 MPa, E=21.3 GPaand ν=0.2. The result of the numerical calculation(Fig. 4) shows that if more than 10,000 elements areused, both principal stress and strain converge. Ifit is greater than 15,000, no matter what meshing

methods we choose, the result is almost the same.The maximum deviation of the principal stress orprincipal strain is within ±0.5%. That means the pro-posed mesh method is very precise. Therefore, a to-tal element number over 15,000 is adopted for thefollowing analysis.

Fig. 3 Numerical model of mass concrete

87

86

85

84

83

82

81

80

79

784000 7000 10000 13000

Total number of element

(a)

Max

imun

pri

ncip

le s

tres

s (M

pa)

16000 19000 22000

4.5

4.4

4.3

4.2

4.1

4.04000 7000 10000 13000

Total number of element

(b)

Max

imun

pri

ncip

le s

trai

n (×

10−3

)

16000 19000 22000

Fig. 4 Maximum principal stresses and strains of mass concrete

Top layer

2.5 m 5.4 m

10.83 m

h

0.0 m

2.5 m

Middle layer

Bottom layer

θ

(a) (b)

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IV. PREDICTION OF DRYING SHRINKAGE

There are many empirical equations that havebeen developed for the prediction of shrinkage. Themost common approaches are those of ACI and CEB.The empirical equations recommend by ACI 209(1992), CEB-FIP (1991) and Kim and Lee (1998)were adopted to predict the shrinkage strain insidemass concrete.

1. ACI 209R-92 Method

ACI Committee 209 recommended this empiri-cal equation in 1992 that allows shrinkage to be esti-mated as a function of time of drying and ultimateshrinkage. The shrinkage, (εsh)t, at any time t afterage 7 days for moist cured concrete is given by

(εsh) t = t35 + t

(εsh)u (3)

where (εsh)u is the value of ultimate shrinkage. Theeffect of others properties of concrete are given bycorrection factors:

(εsh)u=780γcp.γλ.γh(γvs).γs.γ.γα×10−6 (4)

where

1.

γcp =

1.2 , tt = 11.1 , tt = 31.0 , tt = 70.93 , tt = 140.86 , tt = 280.75 , tt = 90

tt: initial moist curing days

2.

γλ = 1.40 – 0.010λ , for 40 ≤ λ ≤ 803.00 – 0.030λ , for 80 < λ ≤ 100

λ : relative humidity (%)

3. γvs=1.2 exp(-0.00472 v/s)

v/s: volume to surface ratio

4. γs=0.89+0.00161s

s: slump (mm)

5.

γϕ =0.3 + 0.0024ϕ , for ϕ ≤ 50%0.90 + 0.002ϕ , for ϕ > 50%

ϕ: fine aggregate percentage (%)

6. γc=0.75+0.00061c

c: cement content (kg)

7. γα= 0.95+0.008α

α : air content (%)

2. CEB-FIP 1990 Method

CEB-FIP 1978 prediction formula of dryingshrinkage was later corrected in 1990 to be CEB-FIP1990. The traditional CEB-FIP 1978 tends to neglectsome material parameters; hence, its predictions arerelatively weak. The corrected CEB-FIP 1990,however, considers the material parameters of 28-daycompressive strength and types of cement. The de-tailed prediction formula is as follows:

εcs=εcso.βs(t−ts) (5)

where εcs is the shrinkage strain at time t after dryingcommences at time ts; εcso is the basic shrinkagecoefficient, which depends on relative humidity, typeof cement and compressive strength of concrete. εcso

is given by

εcso=εs.(fcm).βRH (6)

where

εs(fcm) = [160+10βsc(9− fcmfcmo

)]10−6

fcm = compressive strength (MPa)fcmo = 10 MPaβsc = 4, low heat cement

5, normal cement8, high early strength cement

βRH = – 1.55 ⋅ β sRH , 40% ≤ RH ≤ 99%+ 0.25 , RH ≥ 99%

βsRH = 1 – ( RHRH0

)3

RH = relative humidity (%)RH0 = 100%

βs is a function corresponding to the change of shrink-age with time, which depends on specimen dimen-sions, is given by

β s(t – ts) = [(t – ts)/t1

350(h/h 0)2 + (t – ts)/t1

]0.5 (7)

where

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h = 2Ac/u (mm)Ac = cross section (mm2)u = perimeter (mm)t1 = 1 dayh0 = 100mm

3. Kim and Lee Method

The prediction equation proposed by Kim andLee (1998) allows shrinkage to be estimated as a func-tion of time of drying and length of the diffusion path.The shrinkage, ε(y, t), is given by

ε(y, t)=aebλ (8)

where

a : 353×10−6 for w/c 0.65238×10−6 for w/c 0.40

b : -415.4 for w/c 0.65-426.1 for w/c 0.40

λ : y/ ty : length of the diffusion path (m)t : shrinkage hours

All data needed in the prediction formula (Table2) are based on the design information of the

con-tainer. For comparisons, the predictions are car-ried out up to 20 years, which are the current serviceperiod of the container. The results of the predic-tions are shown in Fig. 5. It can be seen that the pre-dictions of the shrinkage strains according to the CEBand Kim-Lee formulas are very close. The result ac-cording to the ACI prediction formula, however, issignificantly different. To obtain a neutral prediction,the results of the three prediction formulas are aver-aged out and the average drying shrinkage strain isshown in Fig. 6. From the results of Figs. 5 and 6,we can get the drying shrinkage strain on any sectionin the container. The top surface (in verticaldirection) of the container (see Fig. 3) was coveredby mortar of over 100 mm. The bottom surface (invertical direction) was built on a continuous founda-tion plate (3 m thick). Therefore, both surfaces areconsidered as closed surfaces and the water diffusionpath in this direction is ignored. The value is thentransformed to equivalent stress for numericalcalculation. The respective equivalent stress is shownin Table 3 while the corresponding element positionand serial numbers are shown in Fig. 7.

V. NUMERICAL ANALYSIS

The drying shrinkage strains, averaged from

Table 2 Data used in prediction formulas

Design strength Fine aggregate Coarse aggregateSlump (mm) Water (kg/m3) Cement (kg/m3)

(kg/cm2)(MPa) (kg/m3) (kg/m3)

210(20.6) 100 185 280 799 1056

Curing time Prediction strengthAir content (%) R. H. (%) Type of cement

(days) (kg/cm2)(MPa)

Type I2 70 14 280(27.5)

Portland cement

500

400

300

200

100

0

ACI 209

CEB-FIP

KIm-Lee

0 250 500 750

Depth from surface (mm)

Shri

nkag

e st

rain

(×1

0−6 )

1000 1250

Fig. 5 Prediction of concrete shrinkage strain at 20 years of age

500

400

300

200

100

00 250 250 500 750

Depth from surface (mm)

Shri

nkag

e st

rain

(×1

0−6 )

1000 1200 1400

Fig. 6 Averaged shrinkage strain

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the results of the three prediction formulas (see Fig.6), were used to predict the drying shrinkage behav-ior of the container. The results of the numericalcalculation from the top layer, the middle layer andthe bottom layer of the numerical model of container(see Fig. 3) are shown in Figs. 8 and 9.

The maximum principal strains caused by dry-ing shrinkage of concrete are shown in Fig. 8. Thefigure shows that mass concrete will shrink after long-term drying shrinkage. In the interior of concrete,the drying shrinkage strain gradient is relatively mildand the principal strains are smaller at about -150 µ.

Table 3 Equivalent stresses at each element

Serial number of element 8 7, 9 6, 10 5, 11 4, 12 3, 13 2, 14 1, 15

Averaged Shrinkage strain63 77 93 111 135 167 215 319

(×10−6) ACI – 209CEB – FIPKim – Lee

Equivalent stress2.25 2.73 3.29 3.96 4.8 5.93 7.64 11.33

(MPa)

Shrinkage strain1 3 7 15 33 72 157 345

(×10−6)ACI-209


(MPa)

Shrinkage strain75 86 101 122 152 201 282 392

(×10−6)CEB-FIP


(MPa)

151413121110987654321

Fig. 7 Arrangement of elements

600

500

400

300

200

100

0

Top

Middle

Bottom

0 0.5 1 1.5

Cross-section (m)

Shri

nkag

e st

rain

(×1

0−6 )

2 2.5

0m 2.5m

Fig. 8 Radial distributions of shrinkage strains at different heights

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Top layer

Empirical tensilestrength

Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Middle layer

Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Bottom layer

Shri

nkag

e st

ress

(M

Pa)

Cross-section (m)

t1σt2σt3σ

Fig. 9 Radial distributions of principal stresses (averaged model)

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Larger strains appeared in the area near the surface.The maximum strain value, -485 µ, is found on thebottom layer of the mass concrete surface. The mainreason for this is the interaction between the dryingshrinkage of the concrete body and the constraint ofthe concrete bottom. The result is in accordance withthe theoretical prediction.

Fig. 9 shows the result of the principal stress ofconcrete caused by drying shrinkage strain. The prin-cipal stress lies between -2.41 MPa and 5.22 MPa.The stresses distributed on the interior sections ofconcrete are compressive stresses. The ones on thesections near surface are tensile stresses. The maxi-mum tensile stress is found in the surface area of thebottom, as it is influenced by the constraint of theboundary condition.

Tensile strength is an important property of con-crete that greatly affects the extent and the size ofcracking in structures. Most literature (Mehta, 1986;Mindess and Young, 1981) holds that the tensile/com-pressive strength ratio depends on the level of com-pressive strength, the higher the compressive strength,the lower the ratio. In general, the ratio of tensilestrength to compressive strength ranges from 0.07 to0.14. In this study, a more conservative value of 10to 12.5% compressive strength is used to predict stressthat may cause cracking on the concrete. Fig. 9 showsthat the tensile stresses distribute on the top layer nearthe surface area exceed the predicted value of crack-ing stress. The tensile stress of the remaining areawill not lead to the occurrence of cracks. From thefigure, we can determine that the depth of cracks atthe top layer of the mass concrete is about 0.2 m. Theprincipal stress results at the middle layer of the con-tainer (Fig. 9) show that the surface area of the con-crete is under tensile conditions and the interior ofthe concrete is under compressive conditions. Thedepth of the cracks is also determined to be about 0.2m. The principal stress at the bottom layer of thecontainer is similar to the distribution at the middlelayer, but the tensile stress of the surface area is littlehigher than those of the middle layer or the top layer.The reason is that it is influenced by the boundaryconditions. The possible crack depth is also the sameas middle layer at 0.2 m.

The drying shrinkage behavior of the container,predicted based on the shrinkage strains calculatedfrom the ACI 209 formula and the CEB-FIP formula(see Fig. 5), are shown in Figs. 10 and 11. The re-sults show that the distribution of principal stress issimilar to the above-mentioned result. The only dif-ference is that the predicted depths of cracks of theACI mode are shallower. The top layer is about 0.2m, the middle layer is about 0.2 m and the bottomlayer is about 0.2 m. On the other hand, the predicteddepths of CEB-FIP mode are deeper. The top layer

is about 0.2 m, the middle layer is about 0.2 m andthe bottom layer is about 0.2 m. The predicted dry-ing shrinkage strain of the ACI 209 formula has agreater gradient near the surface area, but the affecteddepth is shallower. The tensile stress at the surfacearea is larger, which leads to a shallower crack depth.The results of the CEB-FIP mode show a smaller ten-sile stress at the surface area and deeper cracks. Thisis because its shrinkage gradient is milder than thatof ACI 209. Therefore, although the concrete sur-face appears to have lower tensile stress, the affecteddepth is deeper and a deeper crack will be predicted(see Figs. 10 and 11). In summary, the stress distri-bution predicted by the ACI 209 and CEB-FIP for-mulas is similar to the result of analysis from theaverage of the three prediction formulas (see Fig. 9).

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Top layer


Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Middle layer

Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Bottom layer

Shri

nkag

e st

ress

(M

Pa)

Cross-section (m)

t1σ

t2σ

t3σ

Fig. 10 Radial distributions of principal stresses (ACI 209 model)

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Meanwhile, the predicted depths of the cracks areclose.

VI. NONDESTRUCTIVE TEST

A nondestructive test (Impact Echo) was cho-sen for determining the crack depths in the concrete.The test is economical, rapid and does not cause anydamage in the concrete. The cracks of the containermeasured by nondestructive test are shown in Table4. The corresponding positions of the cracks areshown in Fig. 3. The result shows that the cracks onthe container mostly concentrate on the surface areaat an altitude of 4~7 m and crack depth varies from0.07~0.14 m. The numerically predicted crack depthis about 0.2 m. The two results are not the same.

Table 4 Nondestructive test results

Crack widths Location Crack depthsNumber

(mm) (θ, h) (mm)

1 0.2 ( 3°, 6.5 m) 752 0.2 ( 90°, 7.0 m) 1303 0.2 ( 15°, 7.5 m) 1204 0.2 ( 18°, 6.0 m) 1335 0.1 ( 31°, 6.5 m) 1206 0.1 (115°, 5.5 m) 1257 0.1 (128°, 6.5 m) 1158 0.2 (340°, 4.5 m) 1149 0.2 (350°, 4.5 m) 137

This situation might be caused by not considering thereinforcing bars in the analysis model. The depth oftop reinforcing bar in the whole building is 75 mm.The reinforcing bar can share part of the tensile stressand reduce the occurrence of drying shrinkage cracks.Although the reinforcing bar is not considered andthe result cannot reflect a real situation, the predicteddepth of cracks is reliable for the concrete cover re-gion of the container. Comparison of the numericalprediction result and the nondestructive test datashows that the two results are close and acceptable.This verifies that the proposed method is reliable.

VII. CONCLUSIONS

Numerical method is used to simulate the be-havior of drying shrinkage and to predict crack depthsin mass concrete. From the comparison of the nu-merical results and the nondestructive test results, theconclusions can be summarized as follow:1. The numerical calculation result shows that a bet-

ter convergence can be achieved if the aspect ratio(length, width and height) of the element is within4 and the mesh angle of the element is between45° and 135°. The maximum deviation of princi-pal stress or principal strain is within ±0.5%.

2. Based on the general Hook’s law, the calculatedshrinkage strains in different sections of mass con-crete are converted into equivalent stresses. Thestresses are further applied in the numerical modelto predict drying shrinkage behaviors. This ap-proach is proven to be applicable.

3. The stress distribution situations of mass concretepredicted by the ACI 209 and CEB-FIP formulasare similar to the result gained by averaging theresults of the three prediction formulas.

4. The ACI 209 prediction mode showed greater ten-sile stress near the surface area, but the affecteddepth is shallower and it predicted a shallowercrack. The CEB-FIP prediction mode showed

Fig. 11 Radial distributions of principal stresses (CEB-FIP model)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Top layer


Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Middle layer

Shri

nkag

e st

ress

(M

Pa)

8

6

4

2

0

-2

-40 0.5 1 1.5 2 2.5

Bottom layer

Shri

nkag

e st

ress

(M

Pa)

Cross-section (m)

t1σ

t2σ

t3σ

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smaller tensile stress at the surface area and deepercracks.

5. As the prediction model does not consider the ef-fect of reinforcing bars, the predicted crack depthsare larger than those from the measurements.

6. Comparison of the numerical prediction result andthe nondestructive test data shows that the numeri-cal method to predict the shrinkage behavior ofmass concrete is reliable.

ACKNOWLEDGMENTS

The authors would like to thank Professor Y. Linfor his experimental assistance during this project.

NOMENCLATURE

E elastic modulusfc’ compressive strengthβs a function corresponding to the change of

shrinkage with timeε strainε(y, t) the shrinkage estimated as a function of

time of drying and length of the diffusionpath

εcs the shrinkage strain at time t after dryingcommences at time ts

εcso the basic shrinkage coefficient(εsh)t drying shrinkage at any time t after age 7

days for moist cured concrete(εsh)u the value of ultimate shrinkageν Poisson’s ratioσ stress

REFERENCES

ACI Committee 209, 1992, “Prediction of Creep,Shrinkage and Temperature Effects in ConcreteStructure ,” American Concrete Ins t i tu te ,Farmington Hills, MI, USA.

CEB-FIP, 1991, “Model Code for Concrete 1990,”Final Draft, CEB Bulletin D’Information No. 203,pp. 2.27-2.38, pp. 2.43-2.49.

Kim, J. K., and Lee, C. S., 1998, “Prediction of Dif-ferential Drying Shrinkage in Concrete,” Cementand Concrete Research, Vol. 28, No. 7, pp. 985-994.

Lin, Y., Liou, T., and Tsai, W. H., 1999, “Determin-ing the Crack Depth and the Measurement ErrorsUsing Time-of-Flight Diffraction Techniques,”ACI Materials Journal, Vol. 96, No. 2, pp. 190-195.

Lin, Y., and Su, W. C., 1996, “Use of Stress Wavesfor Determining the Depth of Surface-OpeningCracks in Concrete Structures,” ACI MaterialsJournal, Vol. 93, No. 5, pp. 494-505.

Mehta, P. K., 1986, Concrete: Structure, Properties,and Materials, Prentice-Hall, Inc., EnglewoodCliffs, NJ, USA.

Mindess, S., and Young, J. F., 1981, Concrete,Prentice-Hall, Inc., Englewood Cliffs, NJ, USA.

Manuscript Received: Sep. 25, 2002Revision Received: Oct. 17, 2003

and Accepted: Nov. 21, 2003

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Numerical analysis of shrinkage stresses in a mass concrete